Building a steady rest can be done, but it'l lbe very time consuming and its better to go too heavy with it than too light. Be sure to leave some extra metal on the OD, in case it scars it up, you can take a cleanup cut on it when you are finished with the bore.When you build it, you'll want to do the bore first. As for the taper in the powder chamber, it can be done with the compound rest and a long boring bar. You'll just have to go easy with it.
When you are done with the bore, the best way to turn it would be with a pipe center in the bore, and then you can profile the outer diameter however you want. Since the mortar had a pretty good radius on it, you'll need to do it last, because you wont have enough to grip with the chuck.
Since you are gonna build a steady rest, why not build a radius cuitter ? They arent that hard to do and it'll be a tool that will last you forever if you build it right.
Forget the trace idea with the lathe. Tracers that work with metal lathes are hydraulic and arent even in the same league as wood lathe tracers. Even woth those you need a template. Tracers are pretty much obsolete these days and arent worth the trouble unless you plan on doing a whole lot of the same setup.